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# TOP IN APPLIED MATH I MATH 311

Texas A&M

GPA 3.6

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This 8 page Study Guide was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Study Guide belongs to MATH 311 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/226051/math-311-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15

Math 311 102 mm mm slider About the exam The second examination is tomorrow Thursday June 23 Please bring paper or a bluebook to the exam The exam covers everything on the syllabus to date since the first exam There are 10 questions on the exam What are the main topics mm mm slide Imae and null space basis and dimension 1 0 Sample problems a IfA 0 2 3 4 find a basis for the 9 5 image and a basis forthe null space 1 2 t b If B 2 4 5 forwhich values oft does the null 3 6 space of B have dimension 0 1 2 3 c Give an example of a linear transformation f R3 a R3 such that the null space has dimension 1 and the vector 101 is in the image mm mm mm Eienvalues and eienvectors Sample problems a Find the eigenvalues and eigenvectors 1 72 0 of the matrix 0 2 0 0 71 3 b For which values oft if any does the matrix 1 72 0 0 2 0 have three linearly independent eigenvectors 0 71 t c Express the vector 1 0 0 as a linear combination of the 73 30 760 eigenvectors of the matrix 2 720 40 71 714 22 mm mm slat Orthonormal bases Sample problems a Find an orthonormal basis forlR3 in which one of the basis vectors is g 7 b Starting from the functions 1 x and x2 use the GramSchmidt procedure to construct an orthonormal set with respect to the inner product px qx f01pxqx dx c Find an orthonormal basis forJR3 in which two of the basis vectors span the same plane as do the vectors 110 and 101 ma m5 slam Space curves Sample problems a lfft te J2 cost er 511120 find an equation for the line tangent to the curve at the point where t 0 b If ft tt2t3 is there a point on the curve such that the tangent line at that point passes through the origin c lfft 2tt2 1 t3 eitherfind two points on the curve whose tangent vectors are orthogonal or show that no such points exist ma m5 sliden Sample problems a If guv u2 cosvuv2uev find an equation for the tangent plane to the surface at the point where b Ifguv um uvz find an orthonormal basis forJR3 such that two of the basis vectors are tangent to the surface at the point where u 1 and v 2 c Find a 3 x 3 matrix A such that two of its eigenvectors are tangent to the surface defined by guv v sinu 214 e u 3v at the point on the surface whereu0andv0 mm mm slide Directional derivative Sample problems a lffxyz x2xyyz3 find the directional derivative off in the direction of the unit vector aa atthe point 123 b Find the directional derivative of fxyz x23 y2 in the 2 3 4 direction of an eigenvector of the matrix 0 2 71 0 0 5 c If fx yz x cosy ycosx xyz in what direction is the directional derivative maximal at the point 0 0 0 mm mm Shana The derivative matrix ue Sample problems a ffuv uv find the cosv derivative matrix of f x2 xy z b lffxyz xy find the eigenvalues of the 3x 4y 52 derivative matrix at 000 c Give an example of a transformation f R3 4R3 such that the transformation is locally invertible near the point 000 but is not locally invertible near the point 1 1 1 ma m5 shame Chain rule Sample problems a lffxyz x2y2 22 and 19z represent cylindrical coordinates find E wltgtlt2vvgtltgtgltgtltgtm the derivative matrix of the composite functionf cg c Suppose ltugt f is an invertible coordinate 7 transformation in R2 True or false 1 Bx Bu mamas mm Chane of variables in integrals Sample problems a Evaluate fol 0V 1 12 m2 y2dydx b Evaluate B z2 dxdy dz where B is the ball defined by x2 y2 22 g 4 c Use the coordinate transformation u er cosy v ex siny to evaluate the integral fR xu2v2du do where R is the region in the uvplane corresponding to the region in the xy plane defined by 0 g x g 1 and 0 g y g 712 mu 2m New Math 311 503 Spring 2007 Sample problems for Test 1 Solutions Any problem may be altered or replaced by a different one Problem 1 25 pts Let H be the plane in R3 passing through the points 20 0 110 and 7302 Let Z be the line in R3 passing through the point 111 in the direction 222 i Find a parametric representation for the line Z t2 2 2 1 11 Since the line Z passes through the origin 25 712 an equivalent represen tation is 32 2 2 ii Find a parametric representation for the plane 11 Since the plane H contains the points a 200 b 110 and C 730 2 the vectors b7 a 71 10 and C 7a 75 0 2 are parallel to H Clearly b7 a is not parallel to C 7 a Hence a parametric representation t1b 7 a t2C 7 a a 25171 10 t2750 2 200 iii Find an equation for the plane 11 Since the vectors b 7 a 7110 and C 7 a 750 2 are parallel to the plane H their cross product p b 7 a x C 7 a is orthogonal to H We have that 7 15710 p 2 021752J j 1 0 750 71 39 quot1 0quot quot1 1 k2i2j5k225 75 A point X z y z is in the plane H if and only if p X 7 a 0 This is an equation for the plane In coordinate form 2m 7 2 2y 52 0 or 2x 2y 52 4 iv Find the point where the line Z intersects the plane 11 Let X be the point of intersection Then X 25171 1 0 t2750 2 20 0 for some t1 t2 6 R and also X 32 2 2 for some 3 E R It follows that 7751 7 5752 2 28 t1 28 2752 28 Solving this system of linear equations we obtain that 251 49 252 s 29 Hence X 32 2 2 49 49 49 v Find the angle between the line Z and the plane 11 Let 1 denote the angle between the vectors V 22 2 and p 22 5 Then Vp7 222225 7 18 7 3 val V222222 222252 mm mquot Note that 0 lt j lt 7r2 as cos gt gt 0 Since the vector V is parallel to the line Z while the vector p is orthogonal to the plane H the angle between Z and H is equal to cos gt 7139 7139 3 3 7 7 j 7 7 arccos 7 arcs1n 7 2 2 m m Vi Find the distance from the origin to the plane H The plane H can be de ned by the equatiOn 2m 2y 52 4 Hence the distance from a point 07 290720 to H is equal to l2m0 2240 520 7 4 7 l2m0 230 520 7 4 222252 3 39 In particular the distance from the origin to the plane is equal to m39 Problem 2 15 pts Let fx asinz bcosz c Find a b and c so that f0 1 f 0 2 and f 0 3 f z acosm 7 bsinm f z 7a sinz 7 bcosz Therefore f0 b c f 0 a f 0 7b The desired parameters satisfy the system It follows that a 21 73 and c 4 Thus x 2sinz 7 3cosm 4 Problem 3 20 pts Let A 7 Compute the matrices A2 A3 and pA wherepz2x273z1 A27 3 5 3 5 33572 3551 7 71 20 721 721 723172 72511 78 797 1437142147 71 20 3 5 7 7132072 715201 7 743 15 78 79 72 1 7837972 785791 76 749 7 727 771207 35 107710 25 pA2A 3AIi2lt78 7g 372 1gtlt0 1gt7lt710 720 5 72 4 Problem 4 20 pts Let A 4 73 2 Find the inverse matrix A l 73 4 71 First we merge the matrix A with the identity matrix into one 3 by 6 matrix Then we apply elementary row operations to this matrix until the left part becomes the identity matrix To minimize the number of fractional entries we do not follow the standard elimination procedure Add the third row to the rst and second rows Subtract 2 times the second row from the rst row 2 2 3 1 0 1 0 0 1 1 i2 i1 1 1 1 0 1 1 a 1 1 1 0 1 1 i3 4 i1 0 0 1 i3 4 i1 0 0 1 Subtract the rst row from the second row and then add the rst row to the third row 0 0 1 1 72 71 0 0 1 1 72 71 0 0 1 1 72 71 1 1 1 0 1 1 a 1 1 0 71 3 2 a 1 1 0 71 3 2 73 4 71 0 0 1 73 4 71 0 0 1 73 4 0 1 72 0 Add 3 times the second row to the third row 0 0 1 1 72 71 0 0 1 1 72 71 1 1 0 71 3 2 a 1 1 0 71 3 2 73 4 0 1 72 0 0 7 0 72 7 6 Divide the third row by 7 and then subtract it from the second row 0 0 1 1 2 1 0 0 1 1 2 1 0 0 1 1 2 7 1 1 0 71 3 2 a 1 1 0 71 3 2 a 1 0 0 7 2 0 7 0 0 1 0 g 276 010719 10gt 100 H 7 Interchange the rst row with the second row and then interchange the second row with the third row 00117271 1002 1002 1007320011727101071 01019 01019 00117271 Finally the left part of our 3 by 6 matrix is transformed into the identity matrix Therefore the current right side is the inverse matrix of A Thus Problem 5 20 pts Evaluate the following determinants 524 123 100 432 045 050 3 41 006 007 To evaluate the rst determinant7 we convert the matrix to upper triangular form by applying elementary row operations Applying the same row operations as in the solution of Problem 47 we obtain that 5 72 4 2 2 3 2 2 3 0 0 1 0 0 1 4 73 2 4 73 2 1 1 1 1 1 1 1 1 0 73 4 71 73 4 71 73 4 71 73 4 71 3 4 71 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 7 1 1 0 7 1 0 0 77 0 0 1 7 0 1 0 7 73 4 0 0 7 0 0 1 0 0 1 0 0 1 0 0 0 1 A shorter way is to combine row operations with a row expansion Once we get the matrix 0 0 1 1 1 1 3 4 7 it is convenient to expand its determinant by the rst row Thus 5 72 4 0 4 73 2 1 73 4 71 73 11 4 714717377 73 The other two matrices are already upper triangular Therefore 7157735 OU O TOO 1 3 71 0 5 14624 0 0 6 0 OHgtID Bonus Problem 6 25 pts Find the volume of the parallelepiped bounded by the following three pairs of parallel planes in R3 1 xy2 andzy4 2yz3andyz73 320and25 Let H denote the parallelepiped Any vertex of H is the intersection point of some three planes bounding H Let X0 0340213 be the intersection point of the planes z y 2 y z 3 and z 0 Let X1 zly121 be the intersection point of the planes z y 4 y z 3 and z 0 Let X2 244322 be the intersection point of the planes z y 2 y z 73 and z 0 Let X3 3 34323 be the intersection point of the planes z y 2 y z 3 and z 5 The points X0 X1 X2 and X3 are vertices of the parallelepiped H Moreover the segments XOX1 XOXZ and X0X3 are adjacent edges of the parallelepiped The coordinates of the vertices X0 X1 X2 and X3 can be found from the following systems of linear equations 0y027 1y147 2y227 3y327 y02037 y12137 y22237 y32337 200 210 220 235 Solving them we obtain that X0 7130 X1 130 X2 5730 and X3 472 5 Since vectors V1 X1 7 X0 200 V2 X2 7 X0 6 760 and V3 X3 7 X0 5755 are represented by adjacent edges of the parallelepiped H the volume of H is equal to the absolute value of the following determinant 2 0 6 76 0 276 5760 5 75 5 Thus the volume of H is 60 MATH 3117504 Spring 2008 Sample problems for Test 2 Any problem may be altered or replaced by a different one Problem 1 20 pts Determine which of the following subsets of R3 are subspaces Brie y explain i The set 51 of vectors 7y72 E R3 such that myz 0 ii The set 2 of vectors 7y72 E R3 such that z y z 0 iii The set Sg of vectors 7y72 E R3 such that y2 22 0 iv The set S4 of vectors Ly7 z E R3 such that y2 7 22 0 Problem 2 20 pts Let M22R denote the space of 2 by 2 matrices with real entries Consider a linear operator L M22R a M22R given by L z y 1 2 z y z w 3 4 z w 39 Find the matrix of the operator L with respect to the basis 10 01 00 00 00 E2100 E311 0 o 1 Problem 3 30 pts Consider a linear operator f R3 a R37 fx Ax7 where 1 71 72 A 72 1 3 71 0 1 i Find a basis for the image of f ii Find a basis for the null space of f 1 2 0 Problem 4 30 pts Let B 1 1 1 0 2 1 i Find all eigenvalues of the matrix E ii For each eigenvalue of B7 nd an associated eigenvector iii ls there a basis for R3 consisting of eigenvectors of B Explain iv Find a diagonal matrix D and an invertible matrix U such that B UDU l v Find all eigenvalues of the matrix B2 Bonus Problem 5 20 pts Solve the following system of differential equations nd all solutions diim2 dti y7 d diiyz7 dz 72 m yz

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